Barbara König

The problem of determining the (least) fixpoint of (higher-dimensional) functions over the non-negative reals frequently occurs when dealing with systems endowed with a quantitative semantics. We focus on the situation in which the functions of interest are not known precisely but can only be approximated. As a first contribution we generalize an iteration scheme called dampened Mann iteration, recently introduced in the literature. The improved scheme relaxes previous constraints on parameter sequences, allowing learning rates to converge to zero or not converge at all. While seemingly minor, this flexibility is essential to enable the implementation of chaotic iterations, where only a subset of components is updated in each step, allowing to tackle higher-dimensional problems. Additionally, by allowing learning rates to converge to zero, we can relax conditions on the convergence speed of function approximations, making the method more adaptable to various scenarios. We also show that dampened Mann iteration applies immediately to compute the expected payoff in various probabilistic models, including simple stochastic games, not covered by previous work.

Fixpoints are ubiquitous in computer science and when dealing with quantitative semantics and verification one often considers least fixpoints of (higher-dimensional) functions over the non-negative reals. We show how to approximate the least fixpoint of such functions, focusing on the case in which they are not known precisely, but represented by a sequence of approximating functions that converge to them. We concentrate on monotone and non-expansive functions, for which uniqueness of fixpoints is not guaranteed and standard fixpoint iteration schemes might get stuck at a fixpoint that is not the least. Our main contribution is the identification of an iteration scheme, a variation of Mann iteration with a dampening factor, which, under suitable conditions, is shown to guarantee convergence to the least fixpoint of the function of interest. We then argue that these results are relevant in the context of model-based reinforcement learning for Markov decision processes, showing how the proposed iteration scheme instantiates and allows us to derive convergence to the optimal expected return. More generally, we show that our results can be used to iterate to the least fixpoint almost surely for systems where the function of interest can be approximated with given probabilistic error bounds, as it happens for probabilistic systems which can be explored via sampling.

This paper addresses the following verification task: Given a graph transformation system and a class of initial graphs, can we guarantee (non-)reachability of a given other class of graphs that characterizes bad or erroneous states? Both initial and bad states are characterized by nested conditions (having first-order expressive power). Such systems typically have an infinite state space, causing the problem to be undecidable. We use abstract interpretation to obtain a finite approximation of that state space, and employ counter-example guided abstraction refinement to iteratively obtain suitable predicates for automated verification. Although our primary application is the analysis of graph transformation systems, we state our result in the general setting of reactive systems.